Andrew Garfield and Emma Stone Adopt a Dog Together

The pair have fallen hard for a 2-year-old golden retriever they met just before Christmas at an L.A. rescue.

"She was one of the dogs they first saw, but they weren't sure," a source tells PEOPLE of the pup the pair visited last week at the Spot! Dog Rescue of Los Angeles. "They left and they couldn't stop thinking about her, so they came back for her. And it's true love."

The couple of more than a year initially brought the dog home as a foster pet on Saturday, but officially adopted the purebred on Christmas Eve. The pup, who they've named Ren, came from a high-kill Downey, Calif., shelter where she was surrendered by her previous owner. On Monday, Garfield was spotted showing off their adorable addition on a stroll through the Hollywood Hills.

"She won the doggie lottery," adds the source, who describes the already house-trained pup as "a typical golden puppy. Very, very sweet and loving. They totally fell in love with her."